College Calculus: Level II Integration by Parts

In this lecture we are going to talk about Integration by Parts. First, we will introduce the main equation for integration by parts. We are going to see where it come from. The point of integration by parts is that we will be given a hard integral to solve. What we are going to do, is take the given integral and split it up into two parts, a U part and a dV part. Then we will invoke this formula to convert it into UV minus the integral of VdU, and if we do that right, then the second integral that we get will be an easier integral. We can finish the problem by doing that easier integral.

Antibiotic pharmacokinetics An antibiotic tablet is taken and t hours later the concentration in the bloodstream

is C(t)= 3(e^-0.8t -e^-1.2t)

where C is measured in mgymL.

(a) What is the maximum concentration of the antibiotic

and when does it occur?

(b) Calculate integral of 0 to 2 C(t) dt and interpret your result.

(c) Calculate integral of 0 to infinity of C(t) and explain its meaning.

`

0 Cstd dt and explain its meaning.

3 answers

Last reply by: Dr. William MurrayMon Jun 8, 2015 5:08 PM

Post by Joshua Bowenon May 11, 2015

How do we recognize a Intergration by part quickly?

1 answer

Last reply by: Dr. William MurrayMon Aug 18, 2014 11:04 AM

Post by ahmed ahmedon August 16, 2014

Thank you from the bottom of my heart ... you are of a great help and i got A on my calculus-2 summer class... i am so happy... thank you again and again and forever...

1 answer

Last reply by: Dr. William MurrayTue Feb 11, 2014 4:16 PM

Post by Silvia Gonzalezon February 3, 2014

Hi, thank you for taking the time to prepare such good lectures.I am doing the practice exercise integral from 0 to n of (x-1)sinx.dx and I do not understand how you go from step 8 to 9, can you help me?

1 answer

Last reply by: Dr. William MurraySun Nov 3, 2013 6:23 PM

Post by Christian Fischeron November 3, 2013

Hi Professor, Thank you for a great video. There is one thing I don't understand: If u=x then du=dx but if u=x^2 then du=2xTo me it looks like we differentiated x^2 to get du=2x but did nothing to x to get du=dx? if we had differentiated x we would get du=d*1 What is the explanation for that?

Have a great day Christian

2 answers

Last reply by: Dr. William MurrayFri Aug 23, 2013 11:48 AM

Post by Maximillian Lananderon August 17, 2013

I think you made a boo boo. Not sure. If u is x^2 then du surely = x^2dx ?

3 answers

Last reply by: Dr. William MurrayFri Mar 11, 2016 9:41 AM

Post by Ramez Hajelsawion July 7, 2013

That shortcut blew my mind, I went back and redid all the problems that it works for again just to prove it to myself. Fantastic lecture!

1 answer

Last reply by: Dr. William MurrayTue Apr 23, 2013 7:49 PM

Post by Hye Goo Yoonon April 23, 2013

integration of [(3X^2)/7^3]e^-(x/7)^3 dx 0<X<âˆžAt first I have used by parts method but it was a never ending integration.I have found a formula from the table of integrals, it still gave me unfinished integral. How can I effectively solve this kind of integration by parts? Is there any ways for me to easily approach this kind of integration? Since the range is given,Somehow I have ended up learning about Gamma function. I am very confused now.Thank you

1 answer

Last reply by: Dr. William MurrayFri Jan 18, 2013 5:30 PM

Post by Abdelrahman Megahedon January 17, 2013

How come LIATE doesn't work for example 3?

1 answer

Last reply by: Dr. William MurrayFri Jan 18, 2013 5:30 PM

Post by Shehan Gunasekaraon May 16, 2012

WOW!! This is so goood!! you are better than my lecturer

1 answer

Last reply by: Dr. William MurrayFri Jan 18, 2013 5:33 PM

Post by Emudiare Sowhoon May 7, 2012

How do we get the practice problems?

1 answer

Last reply by: Dr. William MurrayFri Jan 18, 2013 5:34 PM

Post by Kathy Hodgeon November 7, 2011

Thanks. Helped me understand and apply this independently, without having to look at aids, for the first time, despite some previous attempts.

3 answers

Last reply by: Dr. William MurrayFri Jan 18, 2013 5:36 PM

Post by Maimouna Loucheon June 7, 2011

Why does it stop and start over? It stops at the "Shortcut: Tabular Integration".

3 answers

Last reply by: Dr. William MurrayFri Jan 18, 2013 5:37 PM

Post by Maimouna Loucheon June 7, 2011

what is the difference between derivatives and integrals?

1 answer

Last reply by: Dr. William MurrayFri Jan 18, 2013 5:38 PM

Post by Anthony Simonon April 20, 2010

This is a great video, i can't wait for those practice problems.

1 answer

Last reply by: Dr. William MurrayFri Jan 18, 2013 5:41 PM

Post by Stephan Maricon March 13, 2010

Excellent video instruction.My only question is whether or not you have access to supplemental problems. This would really help me practice out the problems.

Integration by Parts

Main formula:

Hints and tips:

Integration by parts never solves the integral completely for you. The point is to reduce a difficult integral to an easier one.

If you have a polynomial multiplied by an exponential or trigonometric function, you can use the tabular integration shortcut.

You may have to use parts more than once in the same problem.

Sometimes, after using parts one or more times, you get the original integral reappearing. Then you can set up an equation for the original integral and find the solution algebraically without doing any more integration.

Use the mnemonic LIATE (Logarithmic, Inverse, Algebraic, Trigonometric, Exponential) to know which part of the expression to make the u.

Sometimes you have to make one or more substitution before you get a form that works well for integration by parts.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Integration by Parts

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.